I don't like putting a rug under the table, it's so easier to clean without it. I think the space looks larger without a rug. So I vote no rug.

The table is dark brown and so are the floors. I'd use a rug to frame the table, lighten up the space and pick up the sage green walls.

Yes. You need definition between the floor and the furniture. If you are worried about spills, use and indoor/outdoor rug or faux sisal.